Consider a small, sinusoidally-pulsating hemisphere on an infinite surface or baffle. It pushes on the air at its surface, causing the adjacent air to move along with the surface, and causing sound to radiate. This hemisphere is the approximate, but accurate, model of the cone of a loudspeaker mounted in a sheet of plywood that is driven with a sound signal. The back-and-forth motion of the air has a direction and a speed that is called “particle velocity” because it is the velocity of a small particle, like a dust mote, suspended in the air and moving with it. (“Particle velocity” is not to be confused with the “wave velocity” of sound, which is about 1100 feet per second, very much greater than the particle velocity.) The particle velocity is in the radial direction, in and out from the center of the hemisphere.
The pulsating motion also produces changes in the air pressure, and the particle velocity can be divided into two parts that differ in their relationship to the air pressure. I call these parts sound, and wind. The total particle motion of the air near the speaker surface is the sum of the sound motion and the wind motion, just as your velocity while walking in a moving train is the sum of the train velocity and your walking velocity. In the sound part of the air motion, the pressure and the particle velocity are in phase, and energy is carried away as sound; in the wind part, the pressure is 90° out of phase with the particle velocity and no energy is carried away (and therefore you can't hear it). Another way of saying this is that when the particle velocity is at all out of phase with the pressure, there is wind as well as sound; and when it is 90° out of phase, there is no sound, only wind. The wind is also referred to a “mass loading” because it results in a mass of air pulsating in and out, that affects the speaker like a weight glued to the speaker cone.
From physics I derived that the sound component of the particle velocity is proportional to 1/r, where r is the radius from the center of the pulsating hemisphere, but the wind component is proportional to 1/r2k, where k is the “wave number” of the sound having the same frequency as the frequency of vibration of the hemisphere (or speaker cone). The quantity k is defined as 2π/λ, where λ is the wavelength of the sound. Both r and k should be in the same units (e.g., r in feet and k in 1/feet, r in meters and k in 1/meters, etc.).
An example: at 30 Hz, the wavelength of sound is 36.6 feet and therefore the wave number is 0.17 ft−1. At that frequency, a 12-inch woofer (approximating a theoretical pulsating hemisphere of radius 0.5 ft) produces sound proportional to 2 (i.e., proportional to 1/r) and wind proportional to 23.5 (i.e., proportional to 1/r2k) right next to the speaker (i.e., at a distance of 0.5 feet). The total air motion is 23.6, which is calculated as the square root of [(2)2+(23.5)2]. The two particle velocity components are added “vectorially” this way due to the 90° phase difference, not because of the direction of particle speed is different for the sound and wind; as noted, all the particle motion is in the radial direction, in or out from the center of the hemisphere.
The proportion of air motion that is sound, which I call the “radiating efficiency,” is then 2/23.6 or 0.085 (8.5%). Clearly, when a 12-inch speaker tries to radiate sound at 30 Hz, most of the speaker cone's action is wasted. Because of this inefficiency, a woofer cone must move through a very large displacement, and it creates a good deal of wind, so much so that light objects in front of the speaker cone can be seen to vibrate. But this motion of the air is almost all inaudible. This example illustrates the general rule of physics, that objects much smaller than a wavelength are not good wave radiators.
If the speaker were made larger, then the radius r and the radiating efficiency would increase. For example, if the speaker radius were 5.8 feet instead of 0.5 feet, then the radiating efficiency would be 50% at the same 30-Hz frequency (i.e. air motion of half wind and half sound), instead of 8.5%. But such a large speaker cone is entirely impractical, not only because of its size but because the sound quality deteriorates as speaker cone size increases. Due to decreased stiffness with increasing size, the cone flaps and oscillates instead of moving as a whole, and that causes sound distortion.
But a single large speaker can be approximated with an array of small speakers. If a large plane area were solidly tiled with speakers all moving in phase, then the radiating efficiency for low frequency would be good because the solid tiling is a close approximation to a single large vibrating area. But for this to work, the speakers must be close together. If there were no neighboring speakers, the wind would fall away as 1/r2 with the distance r from the center of the speaker. However, the other speakers prevent the wind from flowing outward, because the winds from neighboring speakers collide.
I studied this by way of the flux of wind passing through a cylindrical surface, of radius R, concentric with the speaker. I determined that the flux through this cylindrical surface is proportional to 1/R. In an array of hexagonally-spaced speakers (set along lines at 120°) the air pushed by each speaker is confined to a hexagonal cell, which is very close to a cylinder. Because of the neighboring speakers, then, virtually all the wind will be confined inside the cylinder (when it collides with the wind from neighboring speakers) and so the flow of piled-up air away from the baffle will be proportional to what would have gone out of the cylinder, i.e. the flux. Therefore, doubling the spacing between speaker centers will roughly halve the wind perpendicular to the baffle surface and therefore halve the radiating efficiency.
That speakers in an area array should be close for improved bass response was discovered experimentally by Doubt and described in his U.S. Pat. No. 2,602,860. Experimenting with various arrays of speakers, Doubt found no improvement in bass response over that of isolated speakers when the speakers were separated by one diameter, and found the most improvement when the speakers were set very close.
Doubt found that a larger array has a better bass response, and stated in his patent that doubling the size of the array improved the bass response by one octave. However, Doubt had no theoretical understanding, had no idea of how to group the speakers, and related the bass radiating efficiency to the number of speakers instead of to the diameter of the array.